The orthogonality dimension of a graph $G$ over $\mathbb{R}$ is the smallest integer $k$ for which one can assign a nonzero $k$-dimensional real vector to each vertex of $G$, such that every two adjacent vertices receive orthogonal vectors. We prove that for every sufficiently large integer $k$, it is $\mathsf{NP}$-hard to decide whether the orthogonality dimension of a given graph over $\mathbb{R}$ is at most $k$ or at least $2^{(1-o(1)) \cdot k/2}$. We further prove such hardness results for the orthogonality dimension over finite fields as well as for the closely related minrank parameter, which is motivated by the index coding problem in information theory. This in particular implies that it is $\mathsf{NP}$-hard to approximate these graph quantities to within any constant factor. Previously, the hardness of approximation was known to hold either assuming certain variants of the Unique Games Conjecture or for approximation factors smaller than $3/2$. The proofs involve the concept of line digraphs and bounds on their orthogonality dimension and on the minrank of their complement.

翻译：图$G$在实数域$\mathbb{R}$上的正交维度定义为最小的整数$k$，使得可以为$G$的每个顶点分配一个非零的$k$维实向量，且任意两个相邻顶点对应的向量相互正交。我们证明：对于每个足够大的整数$k$，判定给定图在实数域上的正交维度至多为$k$还是至少为$2^{(1-o(1)) \cdot k/2}$是$\mathsf{NP}$困难的。进一步地，我们证明了在有限域上的正交维度以及密切相关的最小秩参数（该参数受信息论中索引编码问题的启发）同样具有此类难度结果。这特别意味着，对这些图量化参数进行任意常数因子逼近是$\mathsf{NP}$困难的。此前，此类逼近的困难性要么依赖于唯一游戏猜想的特定变体，要么仅对小于$3/2$的逼近因子成立。证明过程涉及有向线图的概念，以及对其正交维度和其补图的最小秩的上下界分析。